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Theorem opid 3691
Description: The ordered pair 𝐴, 𝐴 in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
Hypothesis
Ref Expression
opid.1 𝐴 ∈ V
Assertion
Ref Expression
opid 𝐴, 𝐴⟩ = {{𝐴}}

Proof of Theorem opid
StepHypRef Expression
1 dfsn2 3509 . . . 4 {𝐴} = {𝐴, 𝐴}
21eqcomi 2119 . . 3 {𝐴, 𝐴} = {𝐴}
32preq2i 3572 . 2 {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
4 opid.1 . . 3 𝐴 ∈ V
54, 4dfop 3672 . 2 𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}}
6 dfsn2 3509 . 2 {{𝐴}} = {{𝐴}, {𝐴}}
73, 5, 63eqtr4i 2146 1 𝐴, 𝐴⟩ = {{𝐴}}
Colors of variables: wff set class
Syntax hints:   = wceq 1314  wcel 1463  Vcvv 2658  {csn 3495  {cpr 3496  cop 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502  df-op 3504
This theorem is referenced by:  dmsnsnsng  4984  funopg  5125
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